# Algebraic Curves and Riemann Surfaces

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1 Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society

2 Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex Charts and Complex Structures 1 Complex Charts 1 Complex Atlases 3 The Definition of a Riemann Surface 4 Real 2-Manifolds 5 The Genus of a Compact Riemann Surface 6 Complex Manifolds 6 Problems I.I 7 2. First Examples of Riemann Surfaces 7 A Remark on Defining Riemann Surfaces 7 The Projective Line 8 Complex Tori 9 Graphs of Holomorphic Functions 10 Smooth Affine Plane Curves 10 Problems Projective Curves 13 The Projective Plane P 2 13 Smooth Projective Plane Curves 14 Higher-Dimensional Projective Spaces 16 Complete Intersections 17 Local Complete Intersections 17 Problems Further Reading 19 Chapter II. Functions and Maps Functions on Riemann Surfaces 21 Holomorphic Functions 21 Singularities of Functions; Meromorphic Functions 23 Laurent Series 25 The Order of a Meromorphic Function at a Point 26

3 x CONTENTS C Functions 27 Harmonic Functions 27 Theorems Inherited from One Complex Variable 28 Problems II Examples of Meromorphic Functions 30 Meromorphic Functions on the Riemann Sphere 30 Meromorphic Functions on the Projective Line 31 Meromorphic Functions on a Complex Torus 33 Meromorphic Functions on Smooth Plane Curves 35 Smooth Projective Curves 36 Problems II Holomorphic Maps Between Riemann Surfaces 38 The Definition of a Holomorphic Map 38 Isomorphisms and Automorphisms 40 Easy Theorems about Holomorphic Maps 40 Meromorphic Functions and Holomorphic Maps to the Riemann Sphere 41 Meromorphic Functions on a Complex Torus, Again 42 Problems II Global Properties of Holomorphic Maps 44 Local Normal Form and Multiplicity 44 The Degree of a Holomorphic Map between Compact Riemann Surfaces 47 The Sum of the Orders of a Meromorphic Function 49 Meromorphic Functions on a Complex Torus, Yet Again 50 The Euler Number of a Compact Surface 50 Hurwitz's Formula 52 Problems II.4 53 Further Reading 54 Chapter III. More Examples of Riemann Surfaces More Elementary Examples of Riemann Surfaces 57 Lines and Conies 57 Glueing Together Riemann Surfaces 59 Hyperelliptic Riemann Surfaces 60 Meromorphic Functions on Hyperelliptic Riemann Surfaces 61 Maps Between Complex Tori 62 Problems III.l Less Elementary Examples of Riemann Surfaces 66 Plugging Holes in Riemann Surfaces 66 Nodes of a Plane Curve 67 Resolving a Node of a Plane Curve 69 The Genus of a Projective Plane Curve with Nodes 69

4 CONTENTS xi Resolving Monomial Singularities 71 Cyclic Coverings of the Line 73 Problems III Group Actions on Riemann Surfaces 75 Finite Group Actions 75 Stabilizer Subgroups 76 The Quotient Riemann Surface 77 Ramification of the Quotient Map 79 Hurwitz's Theorem on Automorphisms 82 Infinite Groups 82 Problems III Monodromy 84 Covering Spaces and the Fundamental Group 84 The Monodromy of a Finite Covering 86 The Monodromy of a Holomorphic Map 87 Coverings via Monodromy Representations 88 Holomorphic Maps via Monodromy Representations 90 Holomorphic Maps to IP' 91 Hyperelliptic Surfaces 92 Problems III Basic Projective Geometry 94 Homogeneous Coordinates and Polynomials 94 Projective Algebraic Sets 95 Linear Subspaces 95 The Ideal of a Projective Algebraic Set 96 Linear Automorphisms and Changing Coordinates 97 Projections 98 Secant and Tangent Lines 99 Projecting Projective Curves 101 Problems III Further Reading 103 Chapter IV. Integration on Riemann Surfaces Differential Forms 105 Holomorphic 1-Forms 105 Meromorphic 1-Forms 106 Defining Meromorphic Functions and Forms with a Formula 107 Using dz and dz 108 C 1-Forms Forms of Type (1,0) and (0,1) 110 C 2-Forms 110 Problems IV Operations on Differential Forms 112

5 xii CONTENTS Multiplication of 1-Forms by Functions 112 Differentials of Functions 113 The Wedge Product of Two 1-Forms 113 Differentiating 1-Forms 114 Pulling Back Differential Forms 114 Some Notation 115 The Poincare and Dolbeault Lemmas 117 Problems IV Integration on a Riemann Surface 118 Paths 118 Integration of 1-Forms Along Paths 119 Chains and Integration Along Chains 120 The Residue of a Meromorphic 1-Form 121 Integration of 2-Forms 122 Stoke's Theorem 123 The Residue Theorem 123 Homotopy 124 Homology 126 Problems IV Further Reading 127 Chapter V. Divisors and Meromorphic Functions Divisors 129 The Definition of a Divisor 129 The Degree of a Divisor on a Compact Riemann Surface 129 The Divisor of a Meromorphic Function: Principal Divisors 130 The Divisor of a Meromorphic 1-Form: Canonical Divisors 131 The Degree of a Canonical Divisor on a Compact Riemann Surface 132 The Boundary Divisor of a Chain 133 The Inverse Image Divisor of a Holomorphic Map 133 The Ramification and Branch Divisor of a Holomorphic Map 134 Intersection Divisors on a Smooth Projective Curve 135 The Partial Ordering on Divisors 136 Problems V.I Linear Equivalence of Divisors 138 The Definition of Linear Equivalence 138 Linear Equivalence for Divisors on the Riemann Sphere 140 Principal Divisors on a Complex Torus 140 The Degree of a Smooth Projective Curve 142 Bezout's Theorem for Smooth Projective Plane Curves 143 Pliicker's Formula 143 Problems V Spaces of Functions and Forms Associated to a Divisor 145

6 CONTENTS xiii The Definition of the Space L(D) 145 Complete Linear Systems of Divisors 147 Isomorphisms between L(D)'s under Linear Equivalence 148 The Definition of the Space L^(D) 148 The Isomorphism between L^~>(D) and L{D + K) 149 Computation of L{D) for the Riemann Sphere 149 Computation of L(D) for a Complex Torus 150 A Bound on the Dimension of L(D) 151 Problems V Divisors and Maps to Projective Space 153 Holomorphic Maps to Projective Space 153 Maps to Projective Space Given By Meromorphic Functions 154 The Linear System of a Holomorphic Map 155 Base Points of Linear Systems 157 The Hyperplane Divisor of a Holomorphic Map to P n 158 Defining a Holomorphic Map via a Linear System 160 Removing the Base Points 160 Criteria for (/>, to be an Embedding 161 The Degree of the Image and of the Map 164 Rational and Elliptic Normal Curves 165 Working Without Coordinates 166 Problems V Further Reading 167 Chapter VI. Algebraic Curves and the Riemann-Roch Theorem Algebraic Curves 169 Separating Points and Tangents 169 Constructing Functions with Specified Laurent Tails 171 The Transcendence Degree of the Function Field M(X) 174 Computing the Function Field M(X) 177 Problems VI Laurent Tail Divisors 178 Definition of Laurent Tail Divisors 178 Mittag-Leffler Problems and H l {D) 180 Comparing H l Spaces 181 The Finite-Dimensionality of H l (D) 182 Problems VI The Riemann-Roch Theorem and Serre Duality 185 The Riemann-Roch Theorem I 185 The Residue Map 186 Serre Duality 188 The Equality of the Three Genera 191 The Riemann-Roch Theorem II 192

7 xiv CONTENTS Problems VI Further Reading 193 Chapter VII. Applications of Riemann-Roch First Applications of Riemann-Roch 195 How Riemann-Roch implies Algebraicity 195 Criterion for a Divisor to be Very Ample 195 Every Algebraic Curve is Projective 196 Curves of Genus Zero are Isomorphic to the Riemann Sphere 196 Curves of Genus One are Cubic Plane Curves 197 Curves of Genus One are Complex Tori 197 Curves of Genus Two are Hyperelliptic 198 Clifford's Theorem 198 The Canonical System is Base-Point-Free 200 The Existence of Meromorphic 1-Forms 200 Problems VII The Canonical Map 203 The Canonical Map for a Curve of Genus at Least Three 203 The Canonical Map for a Hyperelliptic Curve 203 Finding Equations for Smooth Projective Curves 204 Classification of Curves of Genus Three 205 Classification of Curves of Genus Four 206 The Geometric Form of Riemann-Roch 207 Classification of Curves of Genus Five 209 The Space L{D) for a General Divisor 210 A Few Words on Counting Parameters 211 Riemann's Count of 3g 3 Parameters for Curves of Genus g 212 Problems VII The Degree of Projective Curves 216 The Minimal Degree 216 Rational Normal Curves 216 Tangent Hyperplanes 217 Flexes and Bitangents 219 Monodromy of the Hyperplane Divisors 221 The Surjectivity of the Monodromy 222 The General Position Lemma 224 Points Imposing Conditions on Hypersurfaces 225 Castelnuovo's Bound 228 Curves of Maximal Genus 230 Problems VII Inflection Points and Weierstrass Points 233 Gap Numbers and Inflection Points of a Linear System 233 The Wronskian Criterion 234

8 CONTENTS xv Higher-order Differentials 236 The Number of Inflection Points 238 Flex Points of Smooth Plane Curves 241 Weierstrass Points 241 Problems VII Further Reading 245 Chapter VIII. Abel's Theorem Homology, Periods, and the Jacobian 247 The First Homology Group 247 The Standard Identified Polygon 247 Periods of 1-Forms 247 The Jacobian of a Compact Riemann Surface 248 Problems VIII The Abel-Jacobi Map 249 The Abel-Jacobi Map A on X 249 The Extension of A to Divisors 250 Independence of the Base Point 250 Statement of Abel's Theorem 250 Problems VIII Trace Operations 251 The Trace of a Function 251 The Trace of a 1-Form 252 The Residue of a Trace 253 An Algebraic Proof of the Residue Theorem 253 Integration of a Trace 254 Proof of Necessity in Abel's Theorem 255 Problems VIII Proof of Sufficiency in Abel's Theorem 257 Lemmas Concerning Periods 257 The Proof of Sufficiency 260 Riemann's Bilinear Relations 262 The Jacobian and the Picard Group 263 Problems VIII Abel's Theorem for Curves of Genus One 265 The Abel-Jacobi Map is an Embedding 265 Every Curve of Genus One is a Complex Torus 265 The Group Law on a Smooth Projective Plane Cubic 266 Problems VIII Further Reading 267 Chapter IX. Sheaves and Cech Cohomology Presheaves and Sheaves 269

9 xvi CONTENTS Presheaves 269 Examples of Presheaves 269 The Sheaf Axiom 272 Locally Constant Sheaves 273 Skyscraper Sheaves 273 Global Sections on Compact Riemann Surfaces 275 Restriction to an Open Subset 276 Problems IX Sheaf Maps 278 Definition of a Map between Sheaves 278 Inclusion Maps 278 Differentiation Maps 279 Restriction or Evaluation Maps 279 Multiplication Maps 280 Truncation Maps 281 The Exponential Map 281 The Kernel of a Sheaf Map and Onto Sheaf Maps 282 Short Exact Sequences of Sheaves 284 Exact Sequences of Sheaves 286 Sheaf Isomorphisms 286 Using Sheaves to Define the Category 288 Problems IX Cech Cohomology of Sheaves 290 Cech Cochains 291 Cech Cochaih Complexes 291 Cohomology with respect to a Cover 292 Refinements 293 Cech Cohomology Groups 295 The Connecting Homomorphism 297 The Long Exact Sequence of Cohomology 298 Problems IX Cohomology Computations 301 The Vanishing of H 1 for C Sheaves 302 The Vanishing of H l for Skyscraper Sheaves 303 Cohomology of Locally Constant Sheaves 304 The Vanishing of H 2 (X, O x \D\) 304 De Rham Cohomology 305 Dolbeault Cohomology 306 Problems IX Further Reading 308 Chapter X. Algebraic Sheaves 309

10 CONTENTS xvii 1. Algebraic Sheaves of Functions and Forms 309 Algebraic Curves 309 Algebraic Sheaves of Functions 309 Algebraic Sheaves of Forms 310 The Zariski Topology 311 Problems X.I Zariski Cohomology 312 The Vanishing of H l (X Zar, T) for a Constant Sheaf T 313 The Interpretation of H l ( >) 314 GAGA Theorems 315 Further Computations 316 The Zero Mean Theorem 317 The High Road to Abel's Theorem 319 Problems X Further Reading 321 Chapter XI. Invertible Sheaves, Line Bundles, and H Invertible Sheaves 323 Sheaves of O-Modules 323 Definition of an Invertible Sheaf 324 Invertible Sheaves associated to Divisors 325 The Tensor Product of Invertible Sheaves 326 The Inverse of an Invertible Sheaf 328 The Group of Isomorphism Classes of Invertible Sheaves 329 Problems XI Line Bundles 331 The Definition of a Line Bundle 331 The Tautological Line Bundle for a Map to P n 333 Line Bundle Homomorphisms 334 Defining a Line Bundle via Transition Functions 335 The Invertible Sheaf of Regular Sections of a Line Bundle 337 Sections of the Tangent Bundle and Tangent Vector Fields 340 Rational Sections of a Line Bundle 342 The Divisor of a Rational Section 343 Problems XI Avatars of the Picard Group 345 Divisors Modulo Linear Equivalence and Cocycles 345 Invertible Sheaves Modulo Isomorphism 348 Line Bundles Modulo Isomorphism 351 The Jacobian 356 Problems XI H as a Classifying Space 357 Why il\o*) Classifies Invertible Sheaves and Line Bundles 357

11 xviii CONTENTS Locally Trivial Structures 359 A General Principle Regarding H 360 Cyclic Unbranched Coverings 360 Extensions of Invertible Sheaves 361 First-Order Deformations 364 Problems XI Further Reading 369 References 371 Index of Notation 377

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